Transcript Prime numbers

Contents
Multiples, factors and primes
A Factors
A Prime numbers
A Prime factor decomposition
A HCF and LCM
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FACTORS
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Finding factors
A factor is a whole number that divides
exactly into a given number.
Factors come in pairs.
For example, what are the factors of 30?
1 and 30, 2 and 15, 3 and 10, 5 and 6.
So, in order, the factors of 30 are:
1, 2, 3, 5, 6, 10, 15 and 30.
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Factor finder
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Circle and square puzzle
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PRIME NUMBERS
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Sieve of Eratosthenes
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Prime numbers
If a whole number has two, and only two,
factors it is called a prime number.
For example, the number 17 has only two factors, 1 and 17.
Therefore, 17 is a prime number.
The number 1 has only one factor, 1.
Therefore, 1 is not a prime number.
There is only one even prime number. What is it?
2 is the only even prime number.
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Prime numbers
The first 10 prime numbers are:
2
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3
5
7
11
13
17
19
23
29
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Testing for prime numbers
Is 107 a prime number?
We can check whether or not a number is prime by testing for
divisibility by successive numbers.
Is 107 divisible by 2?
The last digit is a 7 so, no.
Is 107 divisible by 3?
The digit sum is 8 so, no.
We don’t need to check for divisibility by 4 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
Is 107 divisible by 5?
The last digit is a 7 so, no.
We don’t need to check for divisibility by 6 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
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Testing for prime numbers
Is 107 a prime number?
We can check whether or not a number is prime by testing for
divisibility by successive numbers.
Is 107 divisible by 7? Dividing by 7 leaves a remainder so no.
We don’t need to check for divisibility by 8 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
We don’t need to check for divisibility by 9 because if 3
doesn’t divide into 107, then no multiple of 3 can divide into it.
We don’t need to check for divisibility by 10 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
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Testing for prime numbers
Is 107 a prime number?
We can check whether or not a number is prime by testing for
divisibility by successive prime numbers.
We don’t
Why
don’tneed
we need
to check
to check
for divisibility
for divisibility
by 11bybecause
11?
we have
found that no number below 10 divides into 107.
That means that any number that multiplied 11 would have to
be bigger than 10.
Since, 10 × 11 is bigger than 107 we can stop here.
107 is a prime number.
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Testing for prime numbers
When we are testing whether or not a number is prime, we
only have to test for divisibility by prime numbers.
We don’t need to check for divisibility by any number bigger
than the square root of the number.
A number is prime if no prime number less than the
square root of the number divides into it.
Also, all prime numbers greater than 5 must end in a 1, 3, 7
or 9.
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An amazing fact
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PRIME
FACTORS
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Prime factors
A prime factor is a factor that is also a prime number.
For example,
What are the factors of 30?
The factors of 30 are:
1
2
3
5
6
10
15
30
The prime factors of 30 are 2, 3, and 5.
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Products of prime factors
2 × 3 × 5 = 30
2 × 2 × 2 × 7 = 56
This can be written as 23 × 7 = 56
3 × 3 × 11 = 99
This can be written as 32 × 11 = 99
Every whole number greater than 1 is either a
prime number or can be written as a product of
two or more prime numbers.
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The prime factor decomposition
When we write a number as a product of prime factors it
is called the prime factor decomposition.
For example,
The prime factor decomposition of 100 is:
100 = 2 × 2 × 5 × 5
= 22 × 52
There are 2 methods of finding the prime factor decomposition
of a number.
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Factor trees
36
4
2
9
2
3
3
36 = 2 × 2 × 3 × 3
= 22 × 32
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Factor trees
36
3
12
4
2
3
2
36 = 2 × 2 × 3 × 3
= 22 × 32
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Factor trees
2100
30
6
2
70
5
3
10
2
7
5
2100 = 2 × 2 × 3 × 5 × 5 × 7
= 22 × 3 × 52 × 7
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Factor trees
780
78
2
10
39
3
5
2
13
780 = 2 × 2 × 3 × 5 × 13
= 22 × 3 × 5 × 13
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Dividing by prime numbers
2
96
2
48
2
24
2
12
2
6
3
3
96 = 2 × 2 × 2 × 2 × 2 × 3
= 25 × 3
1
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Dividing by prime numbers
3
315
3
105
315 = 3 × 3 × 5 × 7
5
35
= 32 × 5 × 7
7
7
1
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Dividing by prime numbers
2
702
3
351
3
117
3
39
13
13
702 = 2 × 3 × 3 × 3 × 13
= 2 × 33 × 13
1
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Common factor diagram
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The highest common factor
The highest common factor (or HCF) of two numbers
is the highest number that is a factor of both numbers.
We can find the highest common factor of two numbers by
writing down all their factors and finding the largest factor in
both lists.
For example,
Factors of 36 are : 1,
2,
3,
4,
6,
Factors of 45 are : 1,
3,
5,
9,
15,
9,
12,
18,
36.
45.
The HCF of 36 and 45 is 9.
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The highest common factor
What is the highest common factor (HCF) of 24 and 30?
The factors of 24 are:
1
2
3
4
6
8
12
24
10
15
30
The factors of 30 are:
1
2
3
5
6
The highest common factor (HCF) of 24 and 30 is 6.
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The highest common factor
We use the highest common factor when cancelling fractions.
For example,
Cancel the fraction 36 .
48
The HCF of 36 and 48 is 12, so we need to divide the
numerator and the denominator by 12.
÷12
36
48
=
3
4
÷12
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Using prime factors to find the HCF and LCM
We can use the prime factor decomposition to find the HCF
and LCM of larger numbers.
For example,
Find the HCF and the LCM of 60 and 294.
2
2
3
5
60
30
15
5
1
60 = 2 × 2 × 3 × 5
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2
3
7
7
294
147
49
7
1
294 = 2 × 3 × 7 × 7
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Using prime factors to find the HCF and LCM
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7
60
294
2
7
2
5
3
7
HCF of 60 and 294 = 2 × 3 = 6
LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940
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Using prime factors to find the HCF and LCM
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